find all the zeros of f(x)=x3-3x2+4x-12
you know that are all factors of 12. A little regrouping shows you have
f(x) = x^3-3x^2+4x-12
= x^2(x-3) + 4(x-3)
= (x^2+4)(x-3)
so, f(x) = 0 at x = 3, ±2i
To find the zeros of the function f(x) = x^3 - 3x^2 + 4x - 12, you need to solve the equation f(x) = 0. Here's how you can do it:
Step 1: Start by setting f(x) equal to zero: x^3 - 3x^2 + 4x - 12 = 0.
Step 2: Rearrange the equation to group similar terms: x^3 - 3x^2 + 4x - 12 = 0.
Step 3: Now, there are several methods you can use to find the zeros. One approach is to use the Rational Root Theorem. According to this theorem, the possible rational roots of a polynomial are of the form p/q, where p is a factor of the constant term (-12 in this case) and q is a factor of the leading coefficient (1 in this case). Possible p values to try are ±1, ±2, ±3, ±4, ±6, and ±12. Possible q values are ±1.
Step 4: Substitute each potential rational root into the equation to check if it satisfies f(x) = 0. For example, substituting x = 1 yields: 1^3 - 3(1)^2 + 4(1) - 12 = 0. Continue this process for all the potential roots.
Step 5: Once you find a root that satisfies f(x) = 0, you can divide the polynomial by (x - root) to obtain a quadratic equation. Solve the resulting quadratic equation to find the other roots.
In the case of f(x) = x^3 - 3x^2 + 4x - 12, by applying the Rational Root Theorem, you can find that x = 3 is one of the roots. Dividing f(x) by (x - 3), you get the quadratic equation x^2 + 4x + 4, which can be factored as (x + 2)(x + 2). Hence, the zeros of f(x) are x = 3 and x = -2 (with multiplicity 2, due to the repeated factor).
So, the zeros of f(x) = x^3 - 3x^2 + 4x - 12 are x = 3 and x = -2 (with multiplicity 2).